3. Calculate the expected utility of a person who has wealth W = $10000, faces a

Question

3. Calculate the expected utility of a person who has wealth W = $10000, faces a potential loss of C = $5000 with probability and has a utility index u(x) over money:

(i) = 0.01 and u(x) = x2;

(ii) = 0.01 and u(x) = x;

(iii) = 0.1 and u(x) = 2x/10000;

(iv) = 0.1 and u(x) = ln(x) with the natural base

(v) = 0.1 and u(x) = log10(x) with the base 10

Which of the above utility indexes exhibit risk aversion? Find the certainty equivalents and risk premia of the gambles in each of the above cases. Explain the similarity between cases (iv) and (v).

4. Suppose that Alice prefers $700 for sure rather than a lottery. Suppose also that she complies with Independence. ($1000 w.p. 80%; $0 w.p. 20%)

(i) Does she necessarily prefer $600 for sure rather than a lottery ($1000 w.p. 70%; $0 w.p. 30%)

(ii) Does she necessarily prefer a lottery( $700 w.p. 50%; $0 w.p. 50%) to ($1000 w.p. 40%; $0 w.p. 60%)

(iii) Does she necessarily prefer a lottery ($800 w.p. 10%; $0 w.p. 90%) to ($1000 w.p. 8%; $0 w.p. 92%)

Explanation / Answer

A concave utility function means risk averse function

and for concavity U'' should be negative

i. u(x) = x^2 u''(x) = 2 which is positive so not risk averse

ii. u(x) = sqrt(x) u''(x) = -1/4x^(3/2) which is negative so risk averse

iii. u(x) = -2-x/10000 u''(x) = 0 again not risk averse

iv. u(x) = ln(x) u''(x) = -1/x^(2) which is negative so risk averse

v. u(x) = log(x) u''(x) = -1/x^(2)ln(10) which is negative so risk averse

(i) = 0.01 and u(x) = x2;

U(x) = .01*5000*5000 + .99*10000*10000 = 99250000

For CE = x^2 = 99250000 so x = 9962.43

Risk premium = 10000 - 9962.43 = 37.57

(ii) = 0.01 and u(x) = x;

U(x) = .01*sqrt(5000) + .99*sqrt(10000) = 99.71

For CE = sqrt(x) = 99.71 so x = 9941.51

Risk premium = 10000 - 9941.51 = 58.49

(iii) = 0.1 and u(x) = 2x/10000;

U(x) = .01*(-2-5000/10000) + .99*(-2-10000/10000) = -2.995

For CE = -2-x/10000 = -2.995 so x = 9950

Risk premium = 10000 - 9950 = 50

(iv) = 0.1 and u(x) = ln(x) with the natural base

U(x) = .01*ln(5000) + .99*ln(10000) = 9.2034

For CE = ln(x) = 9.2034 so x = 9930.92

Risk premium = 10000 - 9930.92 = 69.08

(v) = 0.1 and u(x) = log10(x) with the base 10

U(x) = .01*log(5000) + .99*log(10000) = 3.997

For CE = log(x) = 3.997 so x = 9930.92

Risk premium = 10000 - 9930.92 = 69.08

we see that certainity equivalent and risk premium is same for (iv) and (v)

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